Analyzing integrated-optical inhomogeneous planar waveguides by Galerkin's method: A detailed comparison of two different basis functions

The trigonometric and Hermite-Gaussian basis functions for determining the modal characteristics of inhomogeneous optical waveguides by means of the Galerkin's method are presented and analyzed. The results obtained with each set of basis functions for mode spectra and field distributions are compared with other exact and approximate methods. The merits and problems arising with each set of basis functions are discussed.     

Cutoff frequencies in planar optical waveguides with arbitrary index profiles: An efficient numerical method

A numerical method for obtaining mode cutoffs for planar waveguides with arbitrary index profiles is developed. The method is based on the Galerkin method in which the wave equation for modes at cutoff is converted to a matrix eigenvalue equation using a set of orthogonal basis functions. Due to different boundary conditions, we have identified two separate cases; one, in which the cover and the substrate indices are identical leading to same behavior of the field at cutoff in these two regions and, the other, in which the two indices are different and hence, the field behaves differently. We have accordingly chosen different basis functions for the two cases. The method results in a generalized matrix eigenvalue problem which has been converted to a standard symmetric matrix eigenvalue analytically. The method has been used to obtain mode cutoffs for waveguides with a variety of index profiles. Comparisons with available exact results show that very good accuracies can be obtained with moderate matrix sizes.     

A new solving method of non-uniform dielectric planar waveguide

A new solving method of non-unifrom dielectric planar waveguide is presented in this paper. By applying this method to solving the problem of planar dielectric waveguide with arbitrary refractive index variation, the advantage of this new method is more evident. By dividing the waveguide into proper numbers and thickness of layers, we can transform the problem of non-uniform dielectric planar waveguide into the multilayer dielectric waveguide to calculate its propagation constantβ.     

Calculation of cutoff frequency of arbitrary weak guidance optical waveguides by extended boundary condition method

The extended boundary condition method is applied to both circular and elliptical regions for calculating cutoff frequencies V_c of weakly guiding optical waveguides, which consist of a uniform cladding and a core with arbitrary shape and index profile. The frequency shift formula method is explained by means of the extended boundary condition method on circular regions. Comparing the method with analytic and other rigorous methods, we show that it is indeed a powerful tool. Numerical results confirm that the mode designation and arrangement order in elliptical and rectangular waveguides are, in general, dependent on the aspect ratio.     

A basic study on the sound field analysis of periodic structures by boundary integral equation method

This study deals with the development of the approximate method to analyze the sound field around equally spaced finite obstacles, using the periodic boundary condition. First, on the assumption that the equally spaced finite obstacles are the periodically arranged obstacles, the sound field is analyzed by boundary integral equation method with a Green’s function which satisfies the periodic boundary condition. Furthermore, by comparing these results and the exact solution by using the fundamental solution as Green’s function, the validity of the approximate method is also investigated. Next, in order to evaluate the applicability of the approximate method, the simple formula using some parameters, i.e., the frequency, the period, and the number of obstacles, etc., is proposed. The results of the sound field analysis applied the formula are presented.     

Coupled effects of nano-size,stretching, and slip boundary conditions on nonlinear vibrations of nano-tube conveying fluid by the homotopy analysis method

In this paper, natural frequency and nonlinear response of carbon nano-tube (CNT) conveying fluid based on the coupling of nonlocal theory and von Karman's stretching have been obtained. The homotopy analysis method (HAM) has been used for solving nonlinear differential equation of system and convergence region of approach presented. Effects of mid-plane stretching, nonlocal parameter and their coupling in the model have been investigated. It has been concluded that stretching effect is significant only for higher-amplitude initial excitations and lower beam aspect ratios. Moreover, by including the slip boundary condition, the effect of nano-size flow has been revealed in the nonlinear vibration model. We have concluded that small-size effects of nano-tube and nano-flow have impressed critical velocity of fluid significantly specially for gas fluid. Analytical results obtained from HAM solution show satisfactory agreement with numerical solutions such as Runge–Kutta. Having an analytical approach, we have been able to investigate the unbounded growth of amplitude of vibrations for flow velocities near the critical value. Moreover, by employing the second-order approximation of Galerkin's method, the estimated natural frequency of the first mode is verified. The obtained results would indicate that the effects of higher mode on the first natural frequency are negligible for the doubly-clamped CNT.